Introduction

Comb-like polymers with high side chain densities are called brush-like polymers. Typical brush-like polymers are obtained by the polymerization of macromonomers, which consist of a linear polymer with a polymerizable group at one of the chain ends. The resulting polymer is called a polymacromonomer (PM). It was shown that the main-chain stiffness of neutral PMs in organic solvents is much larger than that of linear polymers [1,2,3,4,5,6]. This enhanced stiffness was theoretically shown to arize from the repulsive interactions among side chains [7,8,9,10,11].

It is known that repulsive interactions between ionic polymer chains in aqueous media are much stronger than those of neutral polymers in organic solvents [12, 13]. Therefore, brush-like polymers with ionic side chains may have much higher stiffnesses in aqueous media. Rühe et al. [14] obtained several kinds of brush-like polymers with polyelectrolyte side chains. They showed that the cross-sectional diameters of quarternized poly(vinylpyridine) brushes and sulfonated polystyrene brushes determined by small-angle X-ray scattering were not very different from those of neutral brushes. However, they did not determine the stiffness parameter, λ−1, of these polymers.

Previously, Kanemaru et al. [15] obtained PM samples with sodium poly(styrene sulfonate) (NaPSS) side chains by sulfonating PMs consisting of polystyrene (PS). They took light scattering and viscosity measurements of the solutions of NaPSS PM samples with the number of monomeric units, ns, in the side chain at 15 and determined the mean-square radii of gyration and intrinsic viscosities in 0.05 M NaCl aqueous solution as functions of the molecular weight. Upon analyzing the data, they determined λ−1 and the diameter, d, of the polymer. It was shown that λ−1 for NaPSS PM is ~7.5 times larger than that for PS PM in toluene with the same ns, while d for the former PM is ~1.1 times larger than that for the latter polymer. These results show that the effect of electrostatic interactions on λ−1 is very strong. On the other hand, the effects on d are not remarkable, confirming the results of Rühe et al. [14].

For further discussions of the electrostatic contributions to λ−1 and d, it may be important to study the dependence of these properties on the side chain length and the salt concentration, Cs. Here, we prepared NaPSS PM samples with ns at ~110. Light scattering measurements were taken to determine the radius of gyration of each sample by changing the salt concentration. By analysis of the data, λ−1 and d were determined as functions of Cs.

Experimental procedure

Sample preparation

The NaPSS PM samples used in this study were obtained by the sulfonation of PS PM with side chains containing 113 (weight average) monomeric units [16]. Sulfonation was carried out by Vink’s method [17] as follows. PS PM (1 g) dissolved in cyclohexane (50 cm3) was slowly poured into conc. H2SO4 (33 cm3) along with dissolved P2O5 (7.3 g) and stirred for 3 h at 40 °C. After 4 h, the solution was cooled to 0 °C, and ice (17 g) was added. At this time, some polymer was precipitated. The solution was transferred into a separable flask. After removing the acid phase, water was added to the flask, and it was shaken to dissolve the precipitated polymer. The water phase was separated from the organic phase, neutralized with NaOH aq., and poured into ethanol to precipitate the polymer. After drying in vacuo, the polymer was dissolved into water, and the solution was dialyzed against pure water for ~10 days. The water was changed every day. Dialysis was finished when the electric conductivity of the water phase no longer changed. Since the obtained samples had rather wide molecular weight distributions, each sample was fractionally precipitated with 1.0 M NaOH aq. as the solvent and ethanol as the precipitant. The central fraction was used for light scattering measurements. The fractionated samples dissolved in aqueous media were passed through a column packed with ion exchanging resin (Amberlite IR120B and Amberlite IRA400, Organo Co Ltd., mixed at 1:1). The obtained acid-type polymer was neutralized by NaOH aq. to the neutral point, taken as pH = 7.6 [18]. All the samples obtained were freeze-dried. The degree of sulfonation, fs, for each sample was calculated from the weight ratio of C and S, as determined from the elemental analysis. The values are summarized in Table 1, along with the number-average to weight-average molecular weight ratios, Mw/Mn, determined by size-exclusion chromatography with the calibration curve for this PM. The molecular weight of the macromonomer, M0, was calculated as 2.36 × 104 from the molecular weight of the macromonomer before sulfonation, considering full substitution.

Table 1 Properties of polymacromonomer samples consisting of sodium Poly(styrene sulfonate)
Full size table

Light scattering measurements

NaPSS PM samples were dissolved in NaCl aq. of a certain salt concentration, Cs. The solution was stirred for several days at room temperature when Cs was lower than 0.2 M and for more than one week when Cs was higher than 0.5 M. We tried to solve the polymer into NaCl aq. with Cs more than 2.0 M, but the polymer was not fully dissolved.

Light scattering measurements were taken by a multiangle light scattering detector (MALS), DAWN EOS (Wyatt Technologies, Inc.) with a laser of 690 nm wavelength. Each solution with a certain polymer concentration was slowly injected into the flow cell after passing through a hydrophilic poly(tetrafluoroethylene) filter with a 0.45 μm pore size. When the detector voltage became stable, the excess from the voltage for the solvent was taken as the excess scattering intensity of the solution. For each sample, five solutions with different polymer concentrations c were prepared for each Cs. To prevent saturation of the voltage, c was kept lower than 2.2 × 10−4 g cm−3 for samples SF110-5 and SF110-6 and lower than 1.0 × 10−4 g cm−3 for the other samples. At these concentration ranges, the concentration dependence of the scattering intensity divided by c was very small. Therefore, the second virial coefficient was not determined in this study.

The scattering intensities were extrapolated to c → 0 using Berry’s square-root plot to obtain (Kc/Rθ)c=0, where K is the optical constant and Rθ is the Rayleigh ratio at the scattering angle θ. The mean-square radii of gyration 〈S2〉 and Mw were determined from the slope and the intercept at k2 = 0, respectively, for the plot of \((Kc/R_\theta )_{c = 0}^{1/2}\) vs k2, where k is the magnitude of the scattering vector.

The specific refractive index increment (dn/dc) of NaPSS sample SF110-4 was determined using Shimadzu differential refractometer at 436 and 546 nm wavelength, λ0, at Cs = 0.05 and 0.20 M. Before the measurements, each solution was dialyzed against an NaCl solution with the same Cs as that used for the preparation of the solution. The results are as follows: 0.191 and 0.190 (cm3 g−1) for Cs = 0.05 and 0.20 M, respectively, at 436 nm and 0.185 and 0.183 (cm3 g−1) for Cs = 0.05 and 0.20 M, respectively, at 546 nm. The values at 690 nm were calculated from the linear relation between (dn/dc) and \(\lambda _0^{ - 2}\), and those at different Cs values were calculated assuming a linear relation between (dn/dc) and Cs.

Results

Light scattering data

Figure 1 shows plots of \((Kc/R_\theta )_{c = 0}^{1/2}\) vs k2 for SF110 samples in 0.05 M NaCl aq. From the intercept at k2 = 0 and the slope of each line fitted to the data, Mw and 〈S2〉 were determined.

Fig. 1
figure 1

Square-root reciprocal excess scattering intensities plotted against the square of the magnitude of the scattering vector for SF110 samples in 0.05 M NaCl aq

Full size image

All the data of 〈S2〉 for the NaPSS PM in different Cs are summarized in Table 2, along with the Mw values. Since the Mw values at different Cs values agreed within ±6.5%, the averaged value for each sample is indicated. The Cs dependence of 〈S2〉 for the five samples is shown in Fig. 2. The figure shows that 〈S2〉 monotonically decreases with increasing Cs, which is consistent with the 〈S2〉 of linear NaPSS in NaCl aq. However, the Cs dependence of NaPSS PM is much weaker than that of linear NaPSS [19].

Table 2 Radii of gyration of polymacromonomers consisting of sodium Poly(styrene sulfonate)
Full size table
Fig. 2
figure 2

Salt concentration dependence of the radius of gyration for SF110 samples (unfilled circles, SF110-1; filled circles, SF110-2; upward triangles, SF110-3; diamonds, SF110-4; squares, SF110-5; downward triangles, SF110-6)

Full size image

Fig. 3 shows 〈S2〉 at Cs = 0.05 M plotted against the degree of polymerization of the macromonomer Nw, which was calculated from Mw/M0. In the figure, data for PS PM with the same side chain length in toluene [16] are also shown. The figure shows that the 〈S2〉 values for NaPSS PM are much larger than those for PS PM when Nw is less than 102. However, the values for the former polymer approach those for the latter polymer at higher Nw values. This is very different from the case for NaPSS PM with shorter side chains. In the cases of NaPSS PM and PS PM with side chains of 15 monomeric units (SF15 and F15, respectively), the 〈S2〉 values for both polymers are close when Nw is less than 102, but the values for SF15 become much larger than those for F15 with increasing Nw [15].

Fig. 3
figure 3

Radii of gyration of SF110 in 0.05 M NaCl aq. (unfilled circles) and F110 in toluene (filled circles) plotted against the weight-averaged degree of polymerization of the main chain

Full size image

Fitting data

In Fig. 4, the molecular dependence of 〈S2〉 at different Cs values is shown by different symbols. The solid lines show the values calculated from the following equation [20, 21]:

$$\langle S^2\rangle = \frac{L}{{6\lambda }} - \frac{1}{{4\lambda ^2}} + \frac{1}{{4\lambda ^3L}} - \frac{1}{{8\lambda ^4L^2}}\left[ {1 - \exp ( - 2\lambda L)} \right] + \frac{{d^2}}{8}$$
(1)

for the wormlike chain with the diameter d. Here, the contour length, L, of the polymer is calculated from [22]

$$L = M_{\mathrm{w}}/M_{\mathrm{L}} + \delta$$
(2)

with the apparent contribution δ of the side chains near the main chain ends to L. Kanemaru et al. [15] reported that the contour length per repeating unit of the main chain, h, is 0.265 for SF15 when determined from the 〈S2〉 data. Using this h and M0, ML is estimated as 8.91 × 104 nm−1. The solid line in the figure shows the calculated values with λ−1, d, and ML, where d and δ are assumed to be equal. Although the agreement between the experimental and calculated values is moderate, we may estimate the wormlike chain parameters from the fitting results.

Fig. 4
figure 4

Molecular weight dependence of the radius of gyration of SF110 at various salt concentrations. Solid lines show the theoretical fitting based on Eq. (1)

Full size image

The obtained parameters are summarized in Table 3.

Table 3 Stiffness parameter and diameter for polymacromonomer SF110
Full size table

Discussion

Diameter of the molecule

The chain diameter d of NaPSS PM at Cs = 0.05 M is plotted against the number of monomeric units in a side chain, ns, in Fig. 5, denoted by the unfilled circles. The figure also includes the data for PS PM in toluene, shown as the filled circles [6, 16, 23]. The values for NaPSS PM are larger than those for PS PM, and the difference becomes larger with increasing ns.

Fig. 5
figure 5

Molecular diameter plotted against the number of monomeric units in a side chain for NaPSS PM in 0.05 M NaCl aq. (unfilled circles) and PS PM in toluene (filled circles). The solid and dashed lines show the end-to-end distance of the wormlike chain model with parameters for NaPSS [26] and that of the full stretching chain, respectively, with a length twice as long as the side chain of PM

Full size image

The diameter of a brush-like polymer molecule may be comparable to the end-to-end distance of the linear polymer with a length twice as long as its side chain. The solid line in the figure represents the values for the end-to-end distance, 〈R2〉1/2, of the NaPSS chain calculated from the following equation:

$$\langle R^2\rangle = \left\langle {R^2} \right\rangle _0\alpha _R^2$$
(3)

where 〈R2〉0 indicates the mean-square end-to-end distance of the unperturbed wormlike chain and αR is the end-to-end distance expansion factor. The quasi-two-parameter scheme [24] with the Domb–Barrett equation [25] was used for the calculation of αR, with the wormlike chain parameters for the NaPSS chain in 0.05 M NaCl aq. of ML = 800 nm, λ−1 = 5.0 nm, and B = 4.0 nm [26]. The solid line is rather close to the values for PS PM. In particular, the value for NaPSS PM with nS ~ 2 × 102 (the present study) is much larger than the values given by the solid line and those for PS PM in toluene. This suggests that the NaPSS side chains of SF110 are much more extended by the repulsive interactions from the surrounding side chains.

The dashed line shows the end-to-end distance of the fully stretched linear PS chain with a length twice as long as the side chain in the zigzag state. The values for NaPSS PM are rather close to this line, showing that the side chains are almost fully stretched by the repulsive interactions among the side chains.

Stiffness parameter

The dependence of λ−1 on ns for NaPSS PM at CS = 0.05 M is shown in Fig. 6, along with the data for PS PM in toluene. The figure shows that the values of λ−1 for both polymers increase with increasing ns, although the dependence of NaPSS PM is much weaker than that of PS PM.

Fig. 6
figure 6

Stiffness parameters for NaPSS PM in 0.05 M NaCl aq. (circles) and PS PM in toluene (triangles) plotted against the number of monomeric units in a side chain. The dashed and dotted lines show the calculated values for brush-like polymers consisting of Gaussian side chains with excluded-volume parameters for NaPSS and PS, respectively

Full size image

The dashed and dotted lines in the figure show the calculated values for brush-like polymers having Gaussian side chains [10] with the excluded-volume parameters for NaPSS [24] and PS [27], respectively, in the corresponding solvents. Although the data for PS PM are close to the dotted line, the ns dependence of the data for NaPSS PM is very different from the dashed line. Since the λ−1 of the brush-like polymers with rod-like side chains have the same ns dependence as that of brush-like polymers with Gaussian side chains [28], the disagreement between the NaPSS data and the dashed line may not be ascribed to the side chain conformation but to the approximation of the short-range interaction.

In Fig. 7, the λ−1 of SF110 is plotted against Cs, where λ−1 gradually decreases with increasing Cs. Here, we consider the following model to calculate the stiffness parameter of the main chain of the brush-like polymer with polyelectrolyte side chains. For simplicity, the side chains are assumed to be straight and connected perpendicularly to the main chain with an equal spacing, h. Each side chain has an array of charges with equal spacing. By Manning’s counterion condensation hypothesis [29], counterions bind to the charged groups on the polymer chain when the charge spacing is less than the Bjerrum length, QB. Because the real charge spacing of our polymer is about one third of the QB, the effective number, \(n_{\mathrm{s}}^\prime\), of the charged groups in a side chain becomes about one third of ns. The left effective charges are assumed to interact with each other according to the Debye–Hückel potential [30]. The free energy per molecule is obtained by summing the potentials for all the pairs of the charged groups. By calculating the difference between the free energy before and after bending the main chain to a certain curvature from the straight state, the stiffness parameter was obtained (see Appendix for details).

Fig. 7
figure 7

The stiffness parameter for SF110 in NaCl aq. plotted against the salt concentration (circles). The solid line is the eye guide and the other lines are the calculated values (see text)

Full size image

The calculated values are shown by the dotted line in Fig. 7. The theory predicts too much strong dependence of λ−1 on Cs. In this theory, we do not consider the local ionic strength in the brush. For brush-like polymers consisting of side chains with ionic groups, the local concentrations of small ions in the brush may be far different from those outside the brush. Since the concentration of the ionic groups of the side chains is higher than that in the outside solution, the ionic strength inside the brush must also be higher than that outside the brush. To calculate this value, the Donnan equilibrium theory [31] may be applicable.

If the chemical potentials of the ionic salt inside and outside of the brush are in equilibrium, we may write

$$m_ + ^\prime m_ - ^\prime \gamma _ \pm ^\prime = m_ + m_ - \gamma _ \pm$$
(4)

Here, m+ and m− mean the concentrations of positive and negative ions in the outside solution, respectively, γ± expresses the mean activity coefficient of the ions, and the prime designation means the value inside the brush. Here, the difference between the activity coefficients inside and outside of the brush is ignored by an approximation. The local concentration of the ionic groups on the side chains, mp, may be calculated by

$$m_{\mathrm{p}} = \frac{{n_{\mathrm{s}}^\prime }}{{\pi (d/2)^2h}}$$
(5)

By the neutrality condition,

$$m_ + ^\prime = m_{\mathrm{p}} + m_ - ^\prime ,\quad m_ + = m_ - ( = C_{\mathrm{s}})$$
(6)

From these equations, \(m_ + ^\prime\), \(m_ - ^\prime\), and ionic strength or the Debye length inside the brush at each Cs were calculated with d = 39 nm (the averaged value) and h = 0.26 nm. Then, we calculated λ−1 by assuming that the charges on the brushes interact in the ionic atmosphere with this ionic strength. The calculated results are shown by the dashed line in Fig. 7. They are much lower than the dotted line for Cs ≲ 0.1 and approach the line at larger Cs value. The calculated values for Cs ≲ 0.05 are almost constant but are about one order smaller than the experimental values. The difference between the calculated and experimental values may be attributed to the interactions without the electrostatic interactions. Such interactions may be denoted as nonelectric interactions. If we express the contribution from the electrostatic interactions and the nonelectric interactions by \(\lambda _{{\mathrm{el}}}^{ - 1}\) and \(\lambda _{{\mathrm{ne}}}^{ - 1}\), respectively, we may write

$$\lambda ^{ - 1} = \lambda _{{\mathrm{ne}}}^{ - 1} + \lambda _{{\mathrm{el}}}^{ - 1}$$
(7)

The dot-dashed line shows the calculated value with \(\lambda _{{\mathrm{el}}}^{ - 1}\) of the long-dashed line using \(\lambda _{{\mathrm{ne}}}^{ - 1} = 150\) nm, which was chosen to allow the calculated values to be close to the experimental values. The figure indicates that the stiffness parameter of this polymer is dominated by the constant term \(\lambda _{{\mathrm{ne}}}^{ - 1}\). We note that the value of \(\lambda _{{\mathrm{ne}}}^{ - 1}\) is close to λ−1 for the polystyrene polymacromonomer with the same side chain length in toluene. However, the agreement may be accidental because the conformations of the side chains before and after sulfonation are very different. The weak dependence of the observed λ−1 on Cs may be due to the changes of the distribution of the ions in the brush with the change of Cs.

We may also calculate \(\lambda _{{\mathrm{el}}}^{ - 1}\) for SF15 at Cs = 0.05 M by a similar procedure. Without considering the Donnan equilibrium between the inside and outside the brush, we obtained \(\lambda _{{\mathrm{el}}}^{ - 1} = 97\) nm. However, if we consider the Donnan equilibrium with d = 5.5 nm, the value is reduced to \(\lambda _{{\mathrm{el}}}^{ - 1} = 0.40\) nm, which is much smaller than the experimental value. For λ−1 of this polymer, the contribution of the nonelectric interaction must also be dominant. For further discussion, data on the salt concentration dependence of λ−1 for NaPSS PM with different side chain lengths may be necessary.

Conclusion

In this paper, we determined the stiffness parameter λ−1 of a brush-like polymer with side chains consisting of ~110 sodium styrene sulfonate monomeric units in aqueous NaCl. The dependence of λ−1 on the salt concentration was much weaker than that for linear sodium poly(styrene sulfonate) and the prediction of the theory considering the Debye–Hückel potential between charges on the side chains. It was shown that the high ionic strength in the brush modifies the ionic atmosphere and suppresses the interaction among charged groups. To obtain a quantitative agreement between the observed and theoretical values, the contribution from the nonelectric interactions among side chains without the electrostatic interactions have to be considered.